Introduction to Symmetry Analysis
The book is an introductory text on symmetry analysis based on Lie group theory. After an interesting historical preface, the author introduces basic tools for symmetry analysis. He shows how dimensional analysis can be applied to deduce some laws of physics. Symmetry analysis is then applied to several special situations. The following two chapters contain a basic introduction to systems of ordinary differential equations, first order partial differential equations and to classical Hamiltonian mechanics. The next chapter contains a precise definition of one-parameter Lie groups; the author also describes basic tools needed later on: Lie series and Lie algebras. The following chapters show how Lie group techniques can be applied to a study of several problems: ordinary differential equations, partial differential equations as well as several special problems in fluid mechanics (boundary layer models, incompressible Navier-Stokes equations, compressible Euler equations and a certain model of turbulence). The calculation of the determining equations of the group is a tedious job. In order to help the reader, the author attached to the book a CD containing a Mathematica-based software for determining these equations as well as a limited tool for solving them. The last part of the book is devoted to Lie-Bäcklund transformations (and applications to conservation laws) and Bäcklund transformations (and applications to the Burgers potential equation and to the Korteweg-de Vries equation). The book contains a large number of solved problems and exercises, which help to understand the theory. It can be recommended as a very nice introductory text for graduate as well as undergraduate students interested in the subject.